## GPS Mythology – part 2## Secondary relativistic effects
The preceding chapter examined claims regarding the necessary
application of Relativity to the Global Positioning System.
It argued that, providing the satellites orbited at the same altitude
and no adjustments were made to take Relativity into account, the
largest error that could accumulate over a 24 hour period would be an
unnoticeable 15cm. “But at 38 microseconds per day, the relativistic offset in the rates of the satellite clocks is so large that, if left uncompensated, it would cause navigational errors that accumulate faster than 10 km per day! GPS accounts for relativity by electronically adjusting the rates of the satellite clocks, and by building mathematical corrections into the computer chips which solve for the user's location. Without the proper application of relativity, GPS would fail in its navigational functions within about 2 minutes.” [1]
Is he kidding? Nothing much is going to accumulate over a two
minute period other than a miniscule error of 0.2mm. The only
thing this paragraph demonstrates is that the majority of Relativity
followers can’t think straight and function by blindly repeating the
words of other followers.
## Adjustments for eccentric orbits
In an ideal world, GPS satellites would be launched so as to sit in
perfectly circular orbits at predetermined altitudes. In this
way the amount of relativistic effect would be the same for every
satellite and have a constant value throughout its orbit.
Hence an identical timing adjustment could be programmed into each
satellite to negate the effects. Such an adjustment would
allow a GPS receiver not have to take relativity into account. “If a clock's orbit is not perfectly circular, gravitational and motional frequency shifts combine to give rise to the so-called eccentricity effect, a periodic shift of the clock’s rate with a period of almost 12 hours and an amplitude proportional to the orbit’s eccentricity. For an eccentricity as small as 1%, these effects integrate to produce a periodic variation of amplitude 28 ns in the elapsed time recorded by the satellite clock. If it is not taken properly into account, the eccentricity effect could cause an unacceptable periodic navigational error of more than 8 m. The below diagram shows where these relativistic effects come into play.
On the right side of the diagram is the perigee where the orbit is
closest, and the left side is the apogee where it is farthest.
(For visual ease the eccentricity is shown greatly exaggerated.)
## Another explanation?
The fact that adjustments need to be made within the receivers would
seem good evidence for relativity. Because, unlike satellite
adjustments, what goes on inside the receiver can be easily confirmed –
perhaps not by users, but easily enough by receiver manufacturers, of
which there are many.
Note that as the satellite moves from the apogee to the perigee
(farthest to nearest point), it is moving toward Earth. So a
signal emitted during this phase would have the satellites’ speed added
to it and move at speed
## Ballistic model analogyTo answer this we need to look at a simpler analogy. Consider this classical mechanics situation:
Here a gun aims at a target at distance
Now we change things a bit. In the above diagram, the gun
moves toward the target at velocity (1)
This tells us that the signal will arrive earlier by a time difference
proportional to distance and approaching velocity. (2) So how would this translate to a real situation? Let’s say a GPS satellite had an eccentricity of 1%. The below graph shows its relative velocity toward Earth.
Here the horizontal scale represents time in hours over a 12 hour
orbit. Hour zero is the perigee (closest) and hour 6 is
the apogee (farthest). The vertical axis represents
velocity toward the Earth. As can be seen, the velocity
reaches a maximum of 38.76m/s on its approach and a minimum of negative
the same amount on its departure.
There is very little change, fluctuating by ±265km or 1% at most.
This shows the position error varies by plus and minus 3.43m.
This is almost half the 8m described in the referenced
article. In fact it turns out to be exactly half the
relativity-calculated amount. (3)
This corresponds to the amount of cumulative time difference due to
special and general relativity. G, M, and A are the
gravitational constant, Earth’s mass, and satellite’s semi-major axis respectively.So what would be the equivalent equation due to changes in signal speed? It works out to this: (4)
Comparing this equation with (3), we see it has the same form but it is
exactly half the amount. Here is what a chart would look like
comparing them for a 2% eccentricity ( As can be seen, the prediction of relativity gives a maximum of 13.73m, while the prediction due to additive velocity gives a maximum of 6.86m. So it would seem that additive velocities can predict half the measured error. But since the receivers are programmed to accommodate for double that amount, does that mean the additive velocity idea is wrong and relativity is correct? Or is the other half adjusted for elsewhere? We’ll get back to that.
## The Sagnac Effect
The other relativistic effect we apparently need to adjust for is the
Sagnac effect. The literature explaining this is somewhat
confusing so let’s ease into it with a couple of quotes: “In the GPS, the Sagnac effect can produce discrepancies amounting to hundreds of nanoseconds. Observers in the nonrotating ECI inertial frame would not see a Sagnac effect. Instead, they would see that receivers are moving while a signal is propagating. Receivers at rest on Earth are moving quite rapidly (465 m/s at the equator) through the ECI frame. Correcting for the Sagnac effect in the Earth-fixed frame is equivalent to correcting for such receiver motion in the ECI frame.” While Wikipedia says: “GPS observation processing must also compensate for the Sagnac effect. The GPS time scale is defined in an inertial system but observations are processed in an Earth-centered, Earth-fixed (co-rotating) system, a system in which simultaneity is not uniquely defined. A coordinate transformation is thus applied to convert from the inertial system to the ECEF system. The resulting signal run time correction has opposite algebraic signs for satellites in the Eastern and Western celestial hemispheres. Ignoring this effect will produce an east-west error on the order of hundreds of nanoseconds, or tens of meters in position.” [4] Based on this we can conclude that the Sagnac Effect: - has to do with the rotation of the Earth.
- is most significant at the equator.
- is very important to compensate for because it can
result in “tens of meters” of position error.
We will now investigate a more detailed source by the same author here [5]. - If a signal was transmitted east along
the equator it would take 0.1337 seconds to get back to its starting
point (this could be done using mirrors or optic fibres for visible
light, or re-transmitters for radio signals). Calculation:
divide the circumference of the equator by the speed of light:
2*pi*6378137/(2.998x10
^{8}) = 0.13367s - But the equator is moving, and during that time the original transmission point would have moved east by 62m. Calculation: multiply that time by equatorial velocity: 0.13367*465.1 = 62.17m
- It would take an additional 207.4ns to
cover that distance. Calculation: divide that distance by the
speed of light: 62.17/(2.998x10
^{8}) = 2.074x10^{-7}s = 207.4ns
Does this diagram resemble any portion of the GPS? Satellites
transmit their signal to Earth and that’s that. There’s no
transmission back to the satellites, or between satellites, and none of
their signals are made to loop the equator. So in what way
does the Sagnac effect have any relevance? Here a satellite rises above the right horizon, passes directly overhead a user, and goes down over the left horizon. As it does so, notice that it spends time moving toward and away from the user. This graph shows the velocity in a straight-line direction, as viewed by the user:
The velocity is zero when the satellite is directly overhead.
But as it approaches, the toward-velocity goes up to 927.9m/s, when
level with the horizon. And as it recedes, the velocity goes negative
by the same amount. “In 1984 GPS satellites 3, 4, 6, and 8 were used in simultaneous common view between three pairs of earth timing centers, to accomplish closure in performing an around-the-world Sagnac experiment. ... The size of the Sagnac correction varied from 240 to 350 ns.”
“240 to 350ns”? Shouldn’t that be “0 to 207ns”? The
effect is supposed to be due to Earth’s rotation, meaning it should be
zero at the poles. And how did it get up to 350ns when
207.4ns represents the maximum possible (at the equator)?
When level with the horizon, the distance is 25,766km. Using
that value in equation (5) gives a timing value of 266ns (207.4ns will
occur when the satellite is 29 degrees above the horizon).
Notice there is a formula at bottom with the words “Sagnac correction”
pointing to one of the terms. This term looks identical in
form to equation (1), where r)._{R}So what is v? In the paragraph above that it says:“… and let the receiver have velocity
Where “ECI frame” means velocity relative to Earth’s centre, which
corresponds to the velocity described in the above diagram. “One can estimate the discrepancies from the approximate synchronization correction: This looks similar to the “Sagnac” effect in equation (6). It then says: “Within the framework of general relativity, however, one coordinate system should be as legitimate as another.”
That’s true, so why not make the receiver the reference frame and
describe “Measurements made by an observer traveling with a moving receiver can just as well be described in another reference frame, by using transformations that relate the two frames. In the special case of two inertial frames in uniform relative motion, these are the familiar Lorentz transformations.” This may be a clue. Transforming time using Lorentz transforms uses the formula: (7)
For values of (8)
The expression on the right-hand side now looks like the
additive-speed-of-light adjustment in (1). It’s possible that
receivers manufacturers are told to make a Lorentz transform as shown
by (7) and this effectively gives them an additive-speed-of-light
adjustment. “Historically, there has been much confusion about properly accounting for relativistic effects. And it is almost impossible to discover how different manufacturers go about it!” If the interface guide had given specifics for handling the Sagnac effect there should not be “much confusion” as manufacturers would go about it the same way. Perhaps manufacturers do their own experiments and end up settling with the Lorentz time transformation as shown in (7).
## The TOPEX experiment
This brings us back to the earlier discussion about relativistic
adjustments for orbital eccentricity. In order to
resolve the question of reference frames, the
It shows the relativistic error for a GPS satellite with 1.5%
eccentricity. For an eccentricity of 0.015, this calculates
to a relativistic error of 10.3 metres, and this closely matches what
is shown in the diagram.
## Conclusions
- There is no reason why GPS receivers should have to adjust for a Sagnac effect. Various ‘official’ calculations for the effect are inconsistent with each other. Whereas actual measurements don’t match the official calculations but instead correspond to an additive speed of light. Information on how to adjust for the Sagnac effect in receivers is strangely absent in the official user guide and gps.gov website.
- The adjustment for orbital eccentricity is quite likely not a relativistic adjustment but an adjustment for a change in signal speed as the satellite moves toward and away from the receiver. The difference between the relativistic and the change-in-signal-speed predictions should be measurable on Earth but for some reason is done in space where it can’t be verified.
[1]
http://physicscentral.com/explore/writers/will.cfm |

Copyright © 2015
Bernard Burchell, all rights reserved.